论文标题
加速Riemannian优化：用代理处理约束几何惩罚的约束
Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties
论文作者
论文摘要
我们提出了一种全球加速的一阶方法，用于优化平滑和（强烈的）地质连接函数，以在一系列的Hadamard歧管中使用。我们达到了与Nesterov的加速梯度下降相同的收敛速率，直至多重几何惩罚和对数因子。 至关重要的是，我们可以强制执行我们定义的紧凑型集合中的方法。事先完全加速的作品\ emph {诉讼}假设其算法的迭代保留在某些预先指定的紧凑型集中，但两种先前的有限适用性的方法除外。对于我们的流形，这解决了[KY22]中关于获得全球通用加速度的空旷问题，而无需迭代，假设会留在可行的集合中。 在我们的解决方案中，我们设计了一个加速的Riemannian不精目的近端算法，即使可以准确访问近端操作员，这也是未知的结果，并且具有独立的兴趣。对于平滑功能，我们表明我们可以在某些直径的Riemannian球中使用一阶方法实现代理步骤，这足以使全局加速优化。
We propose a globally-accelerated, first-order method for the optimization of smooth and (strongly or not) geodesically-convex functions in a wide class of Hadamard manifolds. We achieve the same convergence rates as Nesterov's accelerated gradient descent, up to a multiplicative geometric penalty and log factors. Crucially, we can enforce our method to stay within a compact set we define. Prior fully accelerated works \emph{resort to assuming} that the iterates of their algorithms stay in some pre-specified compact set, except for two previous methods of limited applicability. For our manifolds, this solves the open question in [KY22] about obtaining global general acceleration without iterates assumptively staying in the feasible set. In our solution, we design an accelerated Riemannian inexact proximal point algorithm, which is a result that was unknown even with exact access to the proximal operator, and is of independent interest. For smooth functions, we show we can implement the prox step inexactly with first-order methods in Riemannian balls of certain diameter that is enough for global accelerated optimization.
